No.
You have unknown functions 
and 
. You have four equations. In order to solve the differential equations, you need at least as many equations as you have functions.
and . You have four equations. In order to solve the differential equations, you need at least as many equations as you have functions.Also, it is not clear that the v functions are with respect to t or to some other variable -- it is odd that you would use dot notation for the v functions but use explicit (t) for x_g
